Option Pricing Under Mixed Bi-fractional Brownian Motion

In the field of modern financial mathematics,option,a kind of financial derivatives,has always been a research hotspot for many scholars and experts because of its risk-avoiding performance.As the rationality of option pricing will affect the stability of the financial market,so many insiders continue to explore the key issue of how to reasonably price options.This paper mainly studies the pricing of European options and permanent American lookback options driven by the mixed bi-fractional Brownian motion,The main contents are as follows:Aiming at the problem of European option pricing,a time-varying hybrid bi-fractional Brownian motion with GARCH structure(HbimFBM model)is constructed.The SSE 50ETF options are selected as an example for model verification,and it is found that compared with the traditional BS model and the option pricing model under constant mixed bi-fractional Brownian motion(bimFBM model),the HbimFBM model can better describe the fractal characteristics of financial asset prices such as self-similarity and long-term memory,as well as the distribution characteristics such as ’peak and thick tail’.If the root-mean-square error(RMSE)and the percentage of improved pricing accuracy(PAI)are used as the measurement criteria,it is found that the pricing effect of the HbimFBM model is better than that of BS and bimFBM model.For the pricing problem of permanent American lookback option,this paper proposes a pricing model of permanent American lookback option with dividend under the mixed bi-fraction Brownian motion.First,the partial differential equations and corresponding boundary conditions of the permanent American lookback call and put options are obtained by using theΔ-hedging principle and the call-put parity formula.Secondly,the variable substitution method is used to solve the closed-form solution and the optimal implementation boundary of the price of the permanent American lookback option.Finally,through numerical experiments,it is verified that the closed solution has the property of linear proportional scaling,the influence of Hurst index H,parameters K,and volatility on the option price is analyzed,and the operating results conform to the law of financial market development.